Question

The standard deviation of a data set \( x_1, x_2, \dots, x_6 \) (\( x_i>0 \)) is 2. If \[ \sum_{i=1}^{9} x_i^2 = 360, \] then the mean of the data set is

• A. \( 4 \)
• B. \( 6 \)
• C. \( 8 \)
• D. \( 10 \)
• E. \( 12 \)

Hint:

Variance is computed using both the sum of squares and the square of the mean.

Answer 1

The Correct Option is B

Step 1: Formula for Variance \[ \sigma^2 = \frac{\sum x_i^2}{n} - \left( \frac{\sum x_i}{n} \right)^2 \] Given that standard deviation \( \sigma = 2 \), so variance \( \sigma^2 = 4 \). Also, we know: \[ \sum x_i^2 = 360, \quad n = 9 \] Step 2: Compute the Mean 
Let \( \bar{x} \) be the mean: \[ 4 = \frac{360}{9} - \bar{x}^2 \] \[ 4 = 40 - \bar{x}^2 \] \[ \bar{x}^2 = 36 \] \[ \bar{x} = 6 \] 
Final Answer: \[ \boxed{6} \]